Your on-board converter/charger should re-charge your battery just fine. Depending what model of converter/charger you have will determine how long it will take to re-charge your battery. Consider these brochure comments from Progressive Dynamics on how long it will take to re-charge deep cycle batteries.
"Progressive Dynamics ran this test on the amount of time it took a PD9155 (55-amp) converter/charger set to three different output voltages to recharge a 125 AH (Amp Hour) battery after it was fully discharged to 10.5-volts.
14.4-VOLTS (Boost Mode) – Returned the battery to 90% of full charge in approximately 3-hours. The battery reached full charge in approximately 11 hours.
13.6-VOLTS (Normal Mode) – Required 40-hours to return the battery to 90% of full charge and 78-hours to reach full charge.
13.2-VOLTS (Storage Mode) – Required 60-hours to return the battery to 90% of full charge and 100-hours to reach full charge."
Also consider this chart of DC voltages verses Deep Cycle Battery charge state:
Running your deep cycle battery below its 50% charge state which is only 12.06VDC is not a good thing to do with your deep cycle battery. The life of the deep cycle battery will be very much reduced when this happens.
It is a good idea to always plug the trailer into at least the garage 120VAC receptacle when you park it at home or install a battery disconnect switch to completely open the battery circuit so that the hidden sources of power drains doesn't drain down your battery. Even pulling the negative terminal of the battery when parked would work as well.
I always just plug my parked trailer into a garage 120VAC receptacle using a 10-gauge (10-3) extension cord and a RV30A-15A "dogbone" type 18-inch long adaptor (WALMART). I never had much luck using the small round black type RV30A-15A adaptors as they always started getting hot on me after being used for any length of time.
When camping off the power grid we use our batteries to run an inverter to power up the 120VAC appliances (Except the air conditioner and high powered Microwave) and also run the 12VDC appliances we want to use but have to have enough battery power to make this happen and not run the batteries more than 11.9VDC before the next morning. Then we connect the shore power cable up to the 2KW honda Generator using the RV30A-15A "dogbone" type adaptor and can re-charge our battery bank back up to the 90% charge state in as little as three hours during breakfast every morning. Then we are good to do this all over again for the next day/night battery run.
Hopefully your battery will re-charge back up all ok.
Usually everywhere we go has generator restriction rules on when you can run the generator so having the smart-mode converter/charger thats works (Not the WFCO models) is a big plus for us.